| L(s) = 1 | − 2-s + 3-s − 6-s − 5·7-s − 11-s + 13-s + 5·14-s + 19-s − 5·21-s + 22-s + 5·25-s − 26-s + 31-s − 33-s − 38-s + 39-s + 41-s + 5·42-s + 15·49-s − 5·50-s − 53-s + 57-s + 59-s + 61-s − 62-s + 66-s − 67-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s − 6-s − 5·7-s − 11-s + 13-s + 5·14-s + 19-s − 5·21-s + 22-s + 5·25-s − 26-s + 31-s − 33-s − 38-s + 39-s + 41-s + 5·42-s + 15·49-s − 5·50-s − 53-s + 57-s + 59-s + 61-s − 62-s + 66-s − 67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{5} \cdot 89^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{5} \cdot 89^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1997834465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1997834465\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$ | \( ( 1 + T )^{5} \) |
| 89 | $C_1$ | \( ( 1 + T )^{5} \) |
| good | 2 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 3 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 11 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 13 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 19 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 31 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 41 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 53 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 59 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 61 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 67 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 71 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 79 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 83 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.72282006503847043878554961982, −6.54039806465804473613249533460, −6.43990167405514036880673967273, −6.22260871702016035936757989764, −6.15016914101372924674402305644, −5.86264754777762407651360049825, −5.53834456831679418423823653303, −5.39380786552582016914671326479, −5.36687048897744757774889742307, −4.90261123884824008000653367381, −4.73886371325820868944513334758, −4.20765274986844528628191041001, −4.15358169950578980128851491494, −4.10413332091225282953197920008, −3.48063164960865883828521727312, −3.40378140098362251982506623413, −3.23408293960316541633238104680, −3.05989161765625952508933069409, −2.84328453596455786442223996939, −2.72951293800155405178401859435, −2.49350725281715591374378314178, −2.48060764897739148584914150169, −1.43759853104020651931146666918, −1.05104919715520723734926819245, −0.69147841022504571756795403130,
0.69147841022504571756795403130, 1.05104919715520723734926819245, 1.43759853104020651931146666918, 2.48060764897739148584914150169, 2.49350725281715591374378314178, 2.72951293800155405178401859435, 2.84328453596455786442223996939, 3.05989161765625952508933069409, 3.23408293960316541633238104680, 3.40378140098362251982506623413, 3.48063164960865883828521727312, 4.10413332091225282953197920008, 4.15358169950578980128851491494, 4.20765274986844528628191041001, 4.73886371325820868944513334758, 4.90261123884824008000653367381, 5.36687048897744757774889742307, 5.39380786552582016914671326479, 5.53834456831679418423823653303, 5.86264754777762407651360049825, 6.15016914101372924674402305644, 6.22260871702016035936757989764, 6.43990167405514036880673967273, 6.54039806465804473613249533460, 6.72282006503847043878554961982
